20 Search Results for "Janson, Svante"

Document
DOI: 10.4230/LIPIcs.AofA.2024.1
Fringe Trees for Random Trees with Given Vertex Degrees

Authors: Gabriel Berzunza Ojeda, Cecilia Holmgren, and Svante Janson

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
We prove that the number of fringe subtrees, isomorphic to a given tree, in uniformly random trees with given vertex degrees, asymptotically follows a normal distribution. As an application, we establish the same asymptotic normality for random simply generated trees (conditioned Galton-Watson trees). Our approach relies on an extension of Gao and Wormald’s (2004) theorem to the multivariate setting.
Cite as

Gabriel Berzunza Ojeda, Cecilia Holmgren, and Svante Janson. Fringe Trees for Random Trees with Given Vertex Degrees. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 1:1-1:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{berzunzaojeda_et_al:LIPIcs.AofA.2024.1,
  author =	{Berzunza Ojeda, Gabriel and Holmgren, Cecilia and Janson, Svante},
  title =	{{Fringe Trees for Random Trees with Given Vertex Degrees}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{1:1--1:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.1},
  URN =		{urn:nbn:de:0030-drops-204369},
  doi =		{10.4230/LIPIcs.AofA.2024.1},
  annote =	{Keywords: Conditioned Galton-Watson trees, fringe trees, simply generated trees, uniformly random trees with given vertex degrees}
}
Document
DOI: 10.4230/LIPIcs.AofA.2024.4
Sparsification of Phylogenetic Covariance Matrices of k-Regular Trees

Authors: Sean Svihla and Manuel E. Lladser

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
Consider a tree T = (V,E) with root ∘ and an edge length function 𝓁:E → ℝ_+. The phylogenetic covariance matrix of T is the matrix C with rows and columns indexed by L, the leaf set of T, with entries C(i,j): = ∑_{e ∈ [i∧ j,o]}𝓁(e), for each i,j ∈ L. Recent work [Gorman & Lladser 2023] has shown that the phylogenetic covariance matrix of a large but random binary tree T is significantly sparsified, with overwhelmingly high probability, under a change-of-basis to the so-called Haar-like wavelets of T. Notably, this finding enables manipulating the spectrum of covariance matrices of large binary trees without the necessity to store them in computer memory but instead performing two post-order traversals of the tree [Gorman & Lladser 2023]. Building on the methods of the aforesaid paper, this manuscript further advances their sparsification result to encompass the broader class of k-regular trees, for any given k ≥ 2. This extension is achieved by refining existing asymptotic formulas for the mean and variance of the internal path length of random k-regular trees, utilizing hypergeometric function properties and identities.
Cite as

Sean Svihla and Manuel E. Lladser. Sparsification of Phylogenetic Covariance Matrices of k-Regular Trees. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 4:1-4:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{svihla_et_al:LIPIcs.AofA.2024.4,
  author =	{Svihla, Sean and Lladser, Manuel E.},
  title =	{{Sparsification of Phylogenetic Covariance Matrices of k-Regular Trees}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{4:1--4:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.4},
  URN =		{urn:nbn:de:0030-drops-204399},
  doi =		{10.4230/LIPIcs.AofA.2024.4},
  annote =	{Keywords: cophenetic matrix, Haar-like wavelets, hierarchical data, hypergeometric functions, metagenomics, phylogenetic covariance matrix, sparsification, ultrametric matrix}
}
Document
DOI: 10.4230/LIPIcs.AofA.2024.5
Bit-Array-Based Alternatives to HyperLogLog

Authors: Svante Janson, Jérémie Lumbroso, and Robert Sedgewick

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
We present a family of algorithms for the problem of estimating the number of distinct items in an input stream that are simple to implement and are appropriate for practical applications. Our algorithms are a logical extension of the series of algorithms developed by Flajolet and his coauthors starting in 1983 that culminated in the widely used HyperLogLog algorithm. These algorithms divide the input stream into M substreams and lead to a time-accuracy tradeoff where a constant number of bits per substream are saved to achieve a relative accuracy proportional to 1/√M. Our algorithms use just one or two bits per substream. Their effectiveness is demonstrated by a proof of approximate normality, with explicit expressions for standard errors that inform parameter settings and allow proper quantitative comparisons with other methods. Hypotheses about performance are validated through experiments using a realistic input stream, with the conclusion that our algorithms are more accurate than HyperLogLog when using the same amount of memory, and they use two-thirds as much memory as HyperLogLog to achieve a given accuracy.
Cite as

Svante Janson, Jérémie Lumbroso, and Robert Sedgewick. Bit-Array-Based Alternatives to HyperLogLog. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{janson_et_al:LIPIcs.AofA.2024.5,
  author =	{Janson, Svante and Lumbroso, J\'{e}r\'{e}mie and Sedgewick, Robert},
  title =	{{Bit-Array-Based Alternatives to HyperLogLog}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{5:1--5:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.5},
  URN =		{urn:nbn:de:0030-drops-204402},
  doi =		{10.4230/LIPIcs.AofA.2024.5},
  annote =	{Keywords: Cardinality estimation, sketching, Hyperloglog}
}
Document
DOI: 10.4230/LIPIcs.AofA.2024.6
Phase Transition for Tree-Rooted Maps

Authors: Marie Albenque, Éric Fusy, and Zéphyr Salvy

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
We introduce a model of tree-rooted planar maps weighted by their number of 2-connected blocks. We study its enumerative properties and prove that it undergoes a phase transition. We give the distribution of the size of the largest 2-connected blocks in the three regimes (subcritical, critical and supercritical) and further establish that the scaling limit is the Brownian Continuum Random Tree in the critical and supercritical regimes, with respective rescalings √{n/log(n)} and √n.
Cite as

Marie Albenque, Éric Fusy, and Zéphyr Salvy. Phase Transition for Tree-Rooted Maps. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{albenque_et_al:LIPIcs.AofA.2024.6,
  author =	{Albenque, Marie and Fusy, \'{E}ric and Salvy, Z\'{e}phyr},
  title =	{{Phase Transition for Tree-Rooted Maps}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{6:1--6:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.6},
  URN =		{urn:nbn:de:0030-drops-204413},
  doi =		{10.4230/LIPIcs.AofA.2024.6},
  annote =	{Keywords: Asymptotic Enumeration, Planar maps, Random trees, Phase transition}
}
Document
DOI: 10.4230/LIPIcs.AofA.2024.9
On Fluctuations of Complexity Measures for the FIND Algorithm

Authors: Jasper Ischebeck and Ralph Neininger

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
The FIND algorithm (also called Quickselect) is a fundamental algorithm to select ranks or quantiles within a set of data. It was shown by Grübel and Rösler that the number of key comparisons required by FIND as a process of the quantiles α ∈ [0,1] in a natural probabilistic model converges after normalization in distribution within the càdlàg space D[0,1] endowed with the Skorokhod metric. We show that the process of the residuals in the latter convergence after normalization converges in distribution to a mixture of Gaussian processes in D[0,1] and identify the limit’s conditional covariance functions. A similar result holds for the related algorithm QuickVal. Our method extends to other cost measures such as the number of swaps (key exchanges) required by FIND or cost measures which are based on key comparisons but take into account that the cost of a comparison between two keys may depend on their values, an example being the number of bit comparisons needed to compare keys given by their bit expansions.
Cite as

Jasper Ischebeck and Ralph Neininger. On Fluctuations of Complexity Measures for the FIND Algorithm. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{ischebeck_et_al:LIPIcs.AofA.2024.9,
  author =	{Ischebeck, Jasper and Neininger, Ralph},
  title =	{{On Fluctuations of Complexity Measures for the FIND Algorithm}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{9:1--9:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.9},
  URN =		{urn:nbn:de:0030-drops-204440},
  doi =		{10.4230/LIPIcs.AofA.2024.9},
  annote =	{Keywords: FIND, Quickselect, rank selection, probabilistic analysis of algorithms, weak convergence, functional limit theorem}
}
Document
DOI: 10.4230/LIPIcs.AofA.2024.13
On the Number of Distinct Fringe Subtrees in Binary Search Trees

Authors: Stephan Wagner

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
A fringe subtree of a rooted tree is a subtree that consists of a vertex and all its descendants. The number of distinct fringe subtrees in random trees has been studied by several authors, notably because of its connection to tree compaction algorithms. Here, we obtain a very precise result for binary search trees: it is shown that the number of distinct fringe subtrees in a binary search tree with n leaves is asymptotically equal to (c₁n)/(log n) for a constant c₁ ≈ 2.4071298335, both in expectation and with high probability. This was previously shown to be a lower bound, our main contribution is to prove a matching upper bound. The method is quite general and can also be applied to similar problems for other tree models.
Cite as

Stephan Wagner. On the Number of Distinct Fringe Subtrees in Binary Search Trees. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 13:1-13:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{wagner:LIPIcs.AofA.2024.13,
  author =	{Wagner, Stephan},
  title =	{{On the Number of Distinct Fringe Subtrees in Binary Search Trees}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{13:1--13:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.13},
  URN =		{urn:nbn:de:0030-drops-204482},
  doi =		{10.4230/LIPIcs.AofA.2024.13},
  annote =	{Keywords: Fringe subtrees, binary search trees, tree compression, minimal DAG, asymptotics}
}
Document
DOI: 10.4230/LIPIcs.AofA.2024.14
Early Typical Vertices in Subcritical Random Graphs of Preferential Attachment Type

Authors: Peter Mörters and Nick Schleicher

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
We study the size of the connected component of early typical vertices in a subcritical inhomogeneous random graph with a kernel of preferential attachment type. The principal tools in our analysis are, first, a coupling of the neighbourhood of a typical vertex in the graph to a killed branching random walk and, second, an asymptotic result for the number of particles absorbed at the killing barrier in this branching random walk.
Cite as

Peter Mörters and Nick Schleicher. Early Typical Vertices in Subcritical Random Graphs of Preferential Attachment Type. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 14:1-14:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{morters_et_al:LIPIcs.AofA.2024.14,
  author =	{M\"{o}rters, Peter and Schleicher, Nick},
  title =	{{Early Typical Vertices in Subcritical Random Graphs of Preferential Attachment Type}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{14:1--14:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.14},
  URN =		{urn:nbn:de:0030-drops-204493},
  doi =		{10.4230/LIPIcs.AofA.2024.14},
  annote =	{Keywords: Inhomogeneous random graphs, preferential attachment, networks, subcritical behaviour, size of components, connectivity, coupling, branching random walk, random tree}
}
Document
DOI: 10.4230/LIPIcs.AofA.2024.15
Asymptotics of Relaxed k-Ary Trees

Authors: Manosij Ghosh Dastidar and Michael Wallner

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
A relaxed k-ary tree is an ordered directed acyclic graph with a unique source and sink in which every node has out-degree k. These objects arise in the compression of trees in which some repeated subtrees are factored and repeated appearances are replaced by pointers. We prove an asymptotic theta-result for the number of relaxed k-ary tree with n nodes for n → ∞. This generalizes the previously proved binary case to arbitrary finite arity, and shows that the seldom observed phenomenon of a stretched exponential term e^{c n^{1/3}} appears in all these cases. We also derive the recurrences for compacted k-ary trees in which all subtrees are unique and minimal deterministic finite automata accepting a finite language over a finite alphabet.
Cite as

Manosij Ghosh Dastidar and Michael Wallner. Asymptotics of Relaxed k-Ary Trees. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{ghoshdastidar_et_al:LIPIcs.AofA.2024.15,
  author =	{Ghosh Dastidar, Manosij and Wallner, Michael},
  title =	{{Asymptotics of Relaxed k-Ary Trees}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{15:1--15:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.15},
  URN =		{urn:nbn:de:0030-drops-204506},
  doi =		{10.4230/LIPIcs.AofA.2024.15},
  annote =	{Keywords: Asymptotic enumeration, stretched exponential, Airy function, directed acyclic graph, Dyck paths, compacted trees, minimal automata}
}
Document
DOI: 10.4230/LIPIcs.AofA.2024.24
Analysis of Regular Sequences: Summatory Functions and Divide-And-Conquer Recurrences

Authors: Clemens Heuberger, Daniel Krenn, and Tobias Lechner

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
In the asymptotic analysis of regular sequences as defined by Allouche and Shallit, it is usually advisable to study their summatory function because the original sequence has a too fluctuating behaviour. It might be that the process of taking the summatory function has to be repeated if the sequence is fluctuating too much. In this paper we show that for all regular sequences except for some degenerate cases, repeating this process finitely many times leads to a "nice" asymptotic expansion containing periodic fluctuations whose Fourier coefficients can be computed using the results on the asymptotics of the summatory function of regular sequences by the first two authors of this paper. In a recent paper, Hwang, Janson, and Tsai perform a thorough investigation of divide-and-conquer recurrences. These can be seen as 2-regular sequences. By considering them as the summatory function of their forward difference, the results on the asymptotics of the summatory function of regular sequences become applicable. We thoroughly investigate the case of a polynomial toll function.
Cite as

Clemens Heuberger, Daniel Krenn, and Tobias Lechner. Analysis of Regular Sequences: Summatory Functions and Divide-And-Conquer Recurrences. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{heuberger_et_al:LIPIcs.AofA.2024.24,
  author =	{Heuberger, Clemens and Krenn, Daniel and Lechner, Tobias},
  title =	{{Analysis of Regular Sequences: Summatory Functions and Divide-And-Conquer Recurrences}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{24:1--24:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.24},
  URN =		{urn:nbn:de:0030-drops-204597},
  doi =		{10.4230/LIPIcs.AofA.2024.24},
  annote =	{Keywords: Regular sequence, Divide-and-Conquer Recurrence, Summatory Function, Asymptotic Analysis}
}
Document
DOI: 10.4230/LIPIcs.AofA.2024.25
Patricia’s Bad Distributions

Authors: Louigi Addario-Berry, Pat Morin, and Ralph Neininger

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
The height of a random PATRICIA tree built from independent, identically distributed infinite binary strings with arbitrary diffuse probability distribution μ on {0,1}^ℕ is studied. We show that the expected height grows asymptotically sublinearly in the number of leaves for any such μ, but can be made to exceed any specific sublinear growth rate by choosing μ appropriately.
Cite as

Louigi Addario-Berry, Pat Morin, and Ralph Neininger. Patricia’s Bad Distributions. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 25:1-25:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{addarioberry_et_al:LIPIcs.AofA.2024.25,
  author =	{Addario-Berry, Louigi and Morin, Pat and Neininger, Ralph},
  title =	{{Patricia’s Bad Distributions}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{25:1--25:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.25},
  URN =		{urn:nbn:de:0030-drops-204600},
  doi =		{10.4230/LIPIcs.AofA.2024.25},
  annote =	{Keywords: PATRICIA tree, random tree, height of tree, analysis of algorithms}
}
Document
DOI: 10.4230/LIPIcs.AofA.2024.28
Sharpened Localization of the Trailing Point of the Pareto Record Frontier

Authors: James Allen Fill, Daniel Q. Naiman, and Ao Sun

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
For d ≥ 2 and i.i.d. d-dimensional observations X^{(1)}, X^{(2)}, … with independent Exponential(1) coordinates, we revisit the study by Fill and Naiman (Electron. J. Probab., 25:Paper No. 92, 24 pp., 2020) of the boundary (relative to the closed positive orthant), or "frontier", F_n of the closed Pareto record-setting (RS) region RS_n := {0 ≤ x ∈ R^d: x ⊀ X^(i) for all 1 ≤ i ≤ n} at time n, where 0 ≤ x means that 0 ≤ x_j for 1 ≤ j ≤ d and x ≺ y means that x_j < y_j for 1 ≤ j ≤ d. With x_+ : = ∑_{j = 1}^d x_j = ‖x‖₁, let F_n^- := min{x_+: x ∈ F_n} and F_n^+ : = max{x_+: x ∈ F_n}. Almost surely, there are for each n unique vectors λ_n ∈ F_n and τ_n ∈ F_n such that F_n^+ = (λ_n)_+ and F_n^- = (τ_n)_+; we refer to λ_n and τ_n as the leading and trailing points, respectively, of the frontier. Fill and Naiman provided rather sharp information about the typical and almost sure behavior of F^+, but somewhat crude information about F^-, namely, that for any ε > 0 and c_n → ∞ we have P(F_n^- - ln n ∈ (- (2 + ε) ln ln ln n, c_n)) → 1 (describing typical behavior) and almost surely limsup (F_n^- - ln n)/(ln ln n) ≤ 0 and liminf (F_n^- - ln n)/(ln ln ln n) ∈ [-2, -1]. In this extended abstract we use the theory of generators (minima of F_n) together with the first- and second-moment methods to improve considerably the trailing-point location results to F_n^- - (ln n - ln ln ln n) ⟶P -ln(d - 1) (describing typical behavior) and, for d ≥ 3, almost surely limsup [F_n^- -(ln n - ln ln ln n)] ≤ -ln(d - 2) + ln 2 and liminf [F_n^- -(ln n - ln ln ln n)] ≥ -ln d - ln 2.
Cite as

James Allen Fill, Daniel Q. Naiman, and Ao Sun. Sharpened Localization of the Trailing Point of the Pareto Record Frontier. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 28:1-28:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{fill_et_al:LIPIcs.AofA.2024.28,
  author =	{Fill, James Allen and Naiman, Daniel Q. and Sun, Ao},
  title =	{{Sharpened Localization of the Trailing Point of the Pareto Record Frontier}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{28:1--28:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.28},
  URN =		{urn:nbn:de:0030-drops-204631},
  doi =		{10.4230/LIPIcs.AofA.2024.28},
  annote =	{Keywords: Multivariate records, Pareto records, generators, interior generators, minima, maxima, record-setting region, frontier, current records, boundary-crossing probabilities, first moment method, second moment method, orthants}
}
Document
DOI: 10.4230/LIPIcs.AofA.2024.30
Depth-First Search Performance in Random Digraphs

Authors: Philippe Jacquet and Svante Janson

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
We present an analysis of the depth-first search algorithm in a random digraph model with independent outdegrees having an arbitrary distribution with finite variance. The results include asymptotics for the distribution of the stack index and depths of the search. The search yields a series of trees of finite size before and after the exploration of a giant tree. Our analysis mainly concerns the giant tree. Most results are first order. This analysis proposed by Donald Knuth in his next to appear volume of The Art of Computer Programming gives interesting insight in one of the most elegant and efficient algorithm for graph analysis due to Tarjan.
Cite as

Philippe Jacquet and Svante Janson. Depth-First Search Performance in Random Digraphs. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{jacquet_et_al:LIPIcs.AofA.2024.30,
  author =	{Jacquet, Philippe and Janson, Svante},
  title =	{{Depth-First Search Performance in Random Digraphs}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{30:1--30:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.30},
  URN =		{urn:nbn:de:0030-drops-204655},
  doi =		{10.4230/LIPIcs.AofA.2024.30},
  annote =	{Keywords: Depth First Search, random digraph, Analysis of Algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.112
Testing Spreading Behavior in Networks with Arbitrary Topologies

Authors: Augusto Modanese and Yuichi Yoshida

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Given the full topology of a network, how hard is it to test if it is evolving according to a local rule or is far from doing so? Inspired by the works of Goldreich and Ron (J. ACM, 2017) and Nakar and Ron (ICALP, 2021), we initiate the study of property testing in dynamic environments with arbitrary topologies. Our focus is on the simplest non-trivial rule that can be tested, which corresponds to the 1-BP rule of bootstrap percolation and models a simple spreading behavior: Every "infected" node stays infected forever, and each "healthy" node becomes infected if and only if it has at least one infected neighbor. Our results are subdivided into two main groups: - If we are testing a single time step of evolution, then the query complexity is O(Δ/ε) or Õ(√n/ε) (whichever is smaller), where Δ and n are the maximum degree of a node and the number of vertices in the underlying graph, respectively. We also give lower bounds for both one- and two-sided error testers that match our upper bounds up to Δ = o(√n) and Δ = O(n^{1/3}), respectively. If ε is constant, then the first of these also holds against adaptive testers. - When testing the environment over T time steps, we have two algorithms that need O(Δ^{T-1}/εT) and Õ(|E|/εT) queries, respectively, where E is the set of edges of the underlying graph. All of our algorithms are one-sided error, and all of them are also non-adaptive, with the single exception of the more complex Õ(√n/ε)-query tester for the case T = 2.
Cite as

Augusto Modanese and Yuichi Yoshida. Testing Spreading Behavior in Networks with Arbitrary Topologies. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 112:1-112:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{modanese_et_al:LIPIcs.ICALP.2024.112,
  author =	{Modanese, Augusto and Yoshida, Yuichi},
  title =	{{Testing Spreading Behavior in Networks with Arbitrary Topologies}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{112:1--112:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.112},
  URN =		{urn:nbn:de:0030-drops-202554},
  doi =		{10.4230/LIPIcs.ICALP.2024.112},
  annote =	{Keywords: Property testing, bootstrap percolation, local phenomena, expander graphs}
}
Document
DOI: 10.4230/LIPIcs.AofA.2022.4
A Modification of the Random Cutting Model

Authors: Fabian Burghart

Published in: LIPIcs, Volume 225, 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)

Abstract
We propose a modification to the random destruction of graphs: Given a finite network with a distinguished set of sources and targets, remove (cut) vertices at random, discarding components that do not contain a source node. We investigate the number of cuts required until all targets are removed, and the size of the remaining graph. This model interpolates between the random cutting model going back to Meir and Moon [Meir and Moon, 1970] and site percolation. We prove several general results, including that the size of the remaining graph is a tight family of random variables for compatible sequences of expander-type graphs, and determine limiting distributions complete binary trees.
Cite as

Fabian Burghart. A Modification of the Random Cutting Model. In 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 225, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{burghart:LIPIcs.AofA.2022.4,
  author =	{Burghart, Fabian},
  title =	{{A Modification of the Random Cutting Model}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{4:1--4:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-230-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{225},
  editor =	{Ward, Mark Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2022.4},
  URN =		{urn:nbn:de:0030-drops-160903},
  doi =		{10.4230/LIPIcs.AofA.2022.4},
  annote =	{Keywords: Random cutting model, Random separation of graphs, Percolation}
}
Document
DOI: 10.4230/LIPIcs.AofA.2022.11
Depth-First Search Performance in a Random Digraph with Geometric Degree Distribution

Authors: Philippe Jacquet and Svante Janson

Published in: LIPIcs, Volume 225, 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)

Abstract
We present an analysis of the depth-first search algorithm in a random digraph model with geometric outdegree distribution. We give also some extensions to general outdegree distributions. This problem posed by Donald Knuth in his next to appear volume of The Art of Computer Programming gives interesting insight in one of the most elegant and efficient algorithm for graph analysis due to Tarjan.
Cite as

Philippe Jacquet and Svante Janson. Depth-First Search Performance in a Random Digraph with Geometric Degree Distribution. In 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 225, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{jacquet_et_al:LIPIcs.AofA.2022.11,
  author =	{Jacquet, Philippe and Janson, Svante},
  title =	{{Depth-First Search Performance in a Random Digraph with Geometric Degree Distribution}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{11:1--11:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-230-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{225},
  editor =	{Ward, Mark Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2022.11},
  URN =		{urn:nbn:de:0030-drops-160978},
  doi =		{10.4230/LIPIcs.AofA.2022.11},
  annote =	{Keywords: Combinatorics, Depth-First Search, Random Digraphs}
}
  • Refine by Author
  • 8 Janson, Svante
  • 2 Fill, James Allen
  • 2 Holmgren, Cecilia
  • 2 Jacquet, Philippe
  • 2 Neininger, Ralph
  • Show More...

  • Refine by Classification
  • 3 Mathematics of computing → Generating functions
  • 3 Mathematics of computing → Random graphs
  • 3 Theory of computation → Sorting and searching
  • 2 Mathematics of computing → Enumeration
  • 2 Mathematics of computing → Paths and connectivity problems
  • Show More...

  • Refine by Keyword
  • 2 U-statistics
  • 2 random tree
  • 1 Airy function
  • 1 Analysis of Algorithms
  • 1 Asymptotic Analysis
  • Show More...

  • Refine by Type
  • 20 document

  • Refine by Publication Year
  • 13 2024
  • 3 2018
  • 2 2022
  • 1 2019
  • 1 2020

© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact